This paper is concerned with the mathematical analysis of terminal value problems for a stochastic non-classical diffusion equation, where the source is assumed to be driven by classical and fractional Brownian motions. Our two problems are to study in the sense of well-posedness and ill-posedness meanings. Here, a terminal value problem is a problem of determining the statistical properties of the initial data from the final time data. In the case 0 < β ≤ 1, where β is the fractional order of a Laplace operator, we show that these are well-posed under certain assumptions. We state a definition of ill-posedness and obtain the ill-poseness results for the problems when β > 1. The major analysis tools in this paper are based on properties of stochastic integrals with respect to the fractional Brownian motion.

In this paper we focus on the p-th moment exponential stability of neutral stochastic pantograph differential equations with Markovian switching (NSPDEwMS). By means of the Lyapunov method, we develop some sufficient conditions on the p-th moment exponential stability for NSPDEwMS. We analyze two examples to show the interest of the main results.

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The Ulam-Hyers-Rassias stability for stochastic systems has been studied by many researchers using the Gronwall-type inequalities, but there is no research paper on the Ulam-Hyers-Rassias stability of stochastic functional differential equations via fixed point methods. The main goal of this paper is to investigate the Ulam-Hyers Stability (HUS) and Ulam-Hyers-Rassias Stability (HURS) of stochastic functional differential equations (SFDEs). Under the fixed point methods and the stochastic analysis techniques, the stability results for SFDE are investigated. We analyze two illustrative examples to show the validity of the results.

This paper focuses on the finite-time stability of linear stochastic fractional-order systems with time delay for $\alpha \in (\frac{1}{2},1)$ α ∈ ( 1 2 , 1 ) . Under the generalized Gronwall inequality and stochastic analysis techniques, the finite-time stability of the solution for linear stochastic fractional-order systems with time delay is investigated. We give two illustrative examples to show the interest of the main results.

We prove that, each probability meassure on [Formula: see text], with all moments, is canonically associated with (i) a ∗-Lie algebra; (ii) a complexity index labeled by pairs of natural integers. The measures with complexity index [Formula: see text] consist of two disjoint classes: that of all measures with finite support and the semi-circle-arcsine class (the discussion in Sec. 4.1 motivates this name). The class [Formula: see text] coincides with the [Formula: see text]-measures in the finite support case and includes the semi-circle laws in the infinite support case. In the infinite support case, the class [Formula: see text] includes the arcsine laws, and the class [Formula: see text] appeared in central limit theorems of quantum random walks in the sense of Konno. The classes [Formula: see text], with [Formula: see text], do not seem to be present in the literature. The class [Formula: see text] includes the Gaussian and Poisson measures and the associated ∗-Lie algebra is the Heisenberg algebra. The class [Formula: see text] includes the non-standard (i.e. neither Gaussian nor Poisson) Meixner distributions and the associated ∗-Lie algebra is a central extension of [Formula: see text]. Starting from [Formula: see text], the ∗-Lie algebra associated to the class [Formula: see text] is infinite dimensional and the corresponding classes include the higher powers of the standard Gaussian.

In this paper we investigate the partial practical exponential stability of neutral stochastic functional differential equations with Markovian switching. The main tool used to prove the results is the Lyapunov method. We analyze an illustrative example to show the applicability and interest of the main results.

In this paper, we investigate the partial asymptotic stability (PAS) of neutral pantograph stochastic differential equations with Markovian switching (NPSDEwMSs). The main tools used to show the results are the Lyapunov method and the stochastic calculus techniques. We discuss a numerical example to illustrate our main results.